Subgroups — #20288323

Order 6784.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811359	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811334	C₂	1	2	yes	maximal, complemented
3	#16811320	C₂	1	4	yes	maximal, complemented
4	#16811303	C₂	1	4	yes	maximal, complemented
5	#16811270	C₂	1	4	yes	maximal, complemented
6	#16811258	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811155	C₂	1	4	yes	maximal, complemented
8	#16811130	C₂	1	4	yes	maximal, complemented
9	C₄×#11719945	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₅₃	C₂	1	4	yes	maximal, complemented
11	C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	2	yes	—
13	C₂×D₄×D₅₃	C₂²	1	4	yes	—
14	C₂²×C₄×D₅₃	C₂²	1	4	yes	complemented
15	#13133210	C₂²	1	8	yes	complemented
16	#13133208	C₂²	1	4	yes	complemented
17	#13133204	C₂²	1	4	yes	complemented
18	#13133199	C₂²	1	8	yes	complemented
19	#13133198	C₂²	1	4	yes	complemented
20	#13133198	C₂²	1	4	yes	complemented
21	#13133197	C₂²	1	4	yes	complemented
22	#13133193	C₂²	1	8	yes	complemented
23	#13133188	C₂²	1	4	yes	—
24	#13133188	C₂²	1	8	yes	complemented
25	#13133185	C₂²	1	4	yes	—
26	#13133185	C₂²	1	4	yes	complemented
27	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
28	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
29	C₄×D₂₁₂	C₂²	1	2	yes	—
30	C₄×D₂₁₂	C₂²	1	2	yes	—
31	D₄×Dic₅₃	C₂²	1	4	yes	complemented
32	#13133179	C₂²	1	4	yes	—
33	C₄×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
34	C₄×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
35	C₄×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
36	#13133177	C₂²	1	8	yes	complemented
37	#13133176	C₂²	1	2	yes	complemented
38	#13133176	C₂²	1	4	yes	complemented
39	#13133174	C₂²	1	4	yes	complemented
40	C₄⋊C₄×D₅₃	C₂²	1	4	yes	complemented
41	C₂²⋊C₄×D₅₃	C₂²	1	8	yes	complemented
42	C₄²×D₅₃	C₂²	1	4	yes	complemented
43	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
44	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	#127×C₅₃	C₂²	1	1	yes	characteristic
46	#124×C₅₃	C₂²	1	2	yes	complemented
47	#121×C₅₃	C₂²	1	4	yes	complemented
48	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
49	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₅₃	C₂³	1	4	yes	—
51	#41×C₅₃	—	4	4	no	—
52	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
53	C₂×D₄×C₅₃	C₂³	1	4	yes	—
54	C₂×D₄×C₅₃	C₂³	1	4	yes	—
55	C₂×D₄×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
59	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
61	C₂²×Dic₅₃	C₂³	1	4	yes	—
62	D₄×D₅₃	—	2	16	no	—
63	#11719945	—	2	8	no	—
64	C₂×D₂₁₂	C₂³	1	2	yes	—
65	C₂×C₄×D₅₃	—	4	8	no	—
66	C₂×C₄×D₅₃	C₂³	1	4	yes	—
67	C₂×C₄×D₅₃	C₂³	1	4	yes	—
68	C₂×C₄×D₅₃	C₂³	1	4	yes	—
69	C₂×C₄×D₅₃	C₂³	1	8	yes	—
70	C₂×Dic₁₀₆	C₂³	1	2	yes	—
71	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
72	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
73	C₄²×C₅₃	C₂³	1	2	yes	—
74	C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₅₃	C₂³	1	8	yes	—
77	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
78	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
79	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
80	#11719921	C₂³	1	4	yes	—
81	#11719921	C₂³	1	8	yes	—
82	#11719921	C₂³	1	4	yes	—
83	C₄⋊Dic₅₃	C₂³	1	2	yes	—
84	C₄⋊Dic₅₃	C₂³	1	2	yes	—
85	C₄⋊Dic₅₃	C₂³	1	8	yes	—
86	#11719919	C₂³	1	8	yes	—
87	#11719919	C₂³	1	4	yes	—
88	#11719919	C₂³	1	4	yes	—
89	C₄×Dic₅₃	C₂³	1	4	yes	—
90	C₄×Dic₅₃	C₂³	1	4	yes	—
91	C₄×Dic₅₃	C₂³	1	4	yes	—
92	C₂³×C₅₃	C₂⁴	1	4	yes	—
93	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂²×D₅₃	—	2	16	no	—
95	C₂²×D₅₃	—	4	4	no	—
96	C₂²×D₅₃	C₂⁴	1	4	yes	—
97	Q₈×C₅₃	—	4	2	no	—
98	D₄×C₅₃	—	2	8	no	—
99	D₄×C₅₃	—	4	4	no	—
100	D₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	8	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
105	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊D₅₃	—	2	8	no	—
110	C₂²⋊D₅₃	—	2	16	no	—
111	C₂×Dic₅₃	—	4	4	no	—
112	C₂×Dic₅₃	C₂⁴	1	4	yes	—
113	C₂×Dic₅₃	C₂⁴	1	4	yes	—
114	C₂×Dic₅₃	C₂⁴	1	4	yes	—
115	D₂₁₂	#41	1	8	yes	complemented
116	C₄×D₅₃	—	2	8	no	—
117	C₄×D₅₃	—	2	8	no	—
118	C₄×D₅₃	—	2	8	no	—
119	C₄×D₅₃	—	4	8	no	—
120	Dic₁₀₆	—	2	4	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	8	no	—
123	C₂²×C₅₃	—	2	4	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
126	C₂×D₅₃	—	2	4	no	—
127	C₂×D₅₃	—	2	4	no	—
128	C₂×D₅₃	—	4	16	no	—
129	C₂×D₅₃	C₂×#41	1	8	yes	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	4	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	C₂×#41	1	4	yes	—
134	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
135	Dic₅₃	—	2	4	no	—
136	Dic₅₃	—	2	4	no	—
137	Dic₅₃	—	2	4	no	—
138	#3481	—	53	1	no	maximal, Sylow
139	C₂×C₅₃	—	4	4	no	—
140	C₂×C₅₃	—	4	1	no	—
141	C₂×C₅₃	C₂²×#41	1	2	yes	—
142	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
143	D₅₃	—	2	8	no	—
144	#559	—	53	1	no	—
145	#554	—	53	2	no	—
146	#553	—	53	4	no	—
147	#551	—	53	4	no	—
148	#546	—	53	4	no	—
149	D₄²	—	53	1	no	—
150	#532	D₅₃	1	1	yes	characteristic, complemented
151	#529	—	53	4	no	—
152	#525	—	53	4	no	—
153	C₄×#41	—	53	2	no	—
154	C₂×C₄×D₄	—	53	4	no	—
155	C₅₃	#3481	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	53	2	no	—
157	C₂×#41	C₂×D₅₃	1	2	yes	complemented
158	C₂²×D₄	—	53	4	no	—
159	C₂³×C₄	—	53	4	no	—
160	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
161	#127	C₂×D₅₃	1	1	yes	characteristic
162	#126	—	53	8	no	—
163	#125	—	53	4	no	—
164	#124	—	53	4	no	—
165	#124	C₂×D₅₃	1	2	yes	complemented
166	#123	—	53	4	no	—
167	#123	—	53	4	no	—
168	#123	—	53	8	no	—
169	C₂²⋊Q₈	—	53	8	no	—
170	C₂²⋊Q₈	—	53	4	no	—
171	#121	—	53	8	no	—
172	#121	—	53	4	no	—
173	#121	C₂×D₅₃	1	4	yes	complemented
174	#120	—	53	4	no	—
175	#120	—	53	4	no	—
176	C₄×Q₈	—	53	2	no	—
177	C₄×Q₈	—	53	2	no	—
178	C₄×D₄	—	53	4	no	—
179	C₄×D₄	—	53	8	no	—
180	C₄×D₄	—	53	4	no	—
181	C₄×D₄	—	53	4	no	—
182	C₄×D₄	—	53	4	no	—
183	C₄×D₄	—	53	8	no	—
184	C₄×D₄	—	53	2	no	—
185	C₄×D₄	—	53	2	no	—
186	C₄×D₄	C₂×D₅₃	1	4	yes	complemented
187	#117	—	53	2	no	—
188	#117	—	53	4	no	—
189	#117	—	53	4	no	—
190	C₂×C₄⋊C₄	—	53	4	no	—
191	C₂×C₂²⋊C₄	—	53	8	no	—
192	C₂×C₄²	—	53	4	no	—
193	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
194	C₂⁴	—	53	4	no	—
195	#41	—	4	4	no	—
196	#41	—	106	8	no	—
197	C₂×Q₈	—	53	2	no	—
198	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
199	C₂×D₄	—	53	4	no	—
200	C₂×D₄	—	53	8	no	—
201	C₂×D₄	—	53	2	no	—
202	C₂×D₄	—	106	16	no	—
203	C₂×D₄	C₂²×D₅₃	1	4	yes	—
204	C₂×D₄	C₂²×D₅₃	1	2	yes	—
205	C₂×D₄	C₂²×D₅₃	1	4	yes	—
206	C₂²×C₄	—	53	4	no	—
207	C₂²×C₄	—	53	4	no	—
208	C₂²×C₄	—	53	4	no	—
209	C₂²×C₄	—	53	4	no	—
210	C₂²×C₄	—	53	8	no	—
211	C₂²×C₄	—	212	8	no	—
212	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
213	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
215	C₄⋊C₄	—	53	2	no	—
216	C₄⋊C₄	—	53	8	no	—
217	C₄⋊C₄	—	53	2	no	—
218	C₄⋊C₄	—	53	4	no	—
219	C₄⋊C₄	—	53	4	no	—
220	C₄⋊C₄	—	53	8	no	—
221	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
222	C₂²⋊C₄	—	53	8	no	—
223	C₂²⋊C₄	—	53	4	no	—
224	C₂²⋊C₄	—	53	4	no	—
225	C₂²⋊C₄	—	53	4	no	—
226	C₂²⋊C₄	—	53	4	no	—
227	C₂²⋊C₄	—	53	8	no	—
228	C₂²⋊C₄	—	53	4	no	—
229	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
230	C₄²	—	53	4	no	—
231	C₄²	—	53	4	no	—
232	C₄²	—	53	4	no	—
233	C₄²	C₂²×D₅₃	1	1	yes	characteristic
234	C₄²	C₂²×D₅₃	1	1	yes	characteristic
235	C₄²	C₂²×D₅₃	1	2	yes	—
236	C₂³	—	53	4	no	—
237	C₂³	—	106	16	no	—
238	C₂³	—	212	4	no	—
239	C₂³	C₂³×D₅₃	1	4	yes	—
240	C₂³	C₂³×D₅₃	1	1	yes	characteristic
241	Q₈	—	4	2	no	—
242	Q₈	—	106	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	D₄	—	53	8	no	—
247	D₄	—	106	16	no	—
248	D₄	—	106	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	53	4	no	—
253	C₂×C₄	—	53	4	no	—
254	C₂×C₄	—	53	4	no	—
255	C₂×C₄	—	106	8	no	—
256	C₂×C₄	—	106	8	no	—
257	C₂×C₄	—	106	8	no	—
258	C₂×C₄	—	212	8	no	—
259	C₂×C₄	—	212	4	no	—
260	C₂×C₄	C₂³×D₅₃	1	2	yes	—
261	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂³×D₅₃	1	4	yes	—
263	C₂×C₄	C₂³×D₅₃	1	2	yes	—
264	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	53	8	no	—
270	C₂²	—	106	4	no	—
271	C₂²	—	106	4	no	—
272	C₂²	—	212	16	no	—
273	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	106	4	no	—
278	C₄	—	106	4	no	—
279	C₄	—	106	4	no	—
280	C₄	#41×D₅₃	1	4	yes	—
281	C₄	#13133262	1	2	yes	—
282	C₂	—	4	4	no	—
283	C₂	—	4	1	no	—
284	C₂	—	106	8	no	—
285	C₂	C₂×#41×D₅₃	1	2	yes	—
286	C₂	C₂×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₅₃×#532	C₂
hypercenter	#532	D₅₃
hypocenter	C₅₃	#3481
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #16811359	C₂	semidirect
2. #16811334	C₂	semidirect
3. #16811320	C₂	semidirect
4. #16811303	C₂	semidirect
5. #16811270	C₂	semidirect
6. #16811258	C₂	semidirect
7. #16811155	C₂	semidirect
8. #16811130	C₂	semidirect
9. C₄×#11719945	C₂	semidirect
10. C₄×D₄×D₅₃	C₂	semidirect
11. C₅₃×#532	C₂	semidirect
14. C₂²×C₄×D₅₃	C₂²	semidirect
15. #13133210	C₂²	semidirect
16. #13133208	C₂²	semidirect
17. #13133204	C₂²	semidirect
18. #13133199	C₂²	semidirect
19. #13133198	C₂²	semidirect
20. #13133198	C₂²	semidirect
21. #13133197	C₂²	semidirect
22. #13133193	C₂²	semidirect
24. #13133188	C₂²	semidirect
26. #13133185	C₂²	semidirect
27. C₄×Dic₁₀₆	C₂²	semidirect
28. C₄×Dic₁₀₆	C₂²	semidirect
31. D₄×Dic₅₃	C₂²	semidirect
33. C₄×C₂²⋊D₅₃	C₂²	semidirect
34. C₄×C₂²⋊D₅₃	C₂²	semidirect
35. C₄×C₂²⋊D₅₃	C₂²	semidirect
36. #13133177	C₂²	semidirect
37. #13133176	C₂²	semidirect
38. #13133176	C₂²	semidirect
39. #13133174	C₂²	semidirect
40. C₄⋊C₄×D₅₃	C₂²	semidirect
41. C₂²⋊C₄×D₅₃	C₂²	semidirect
42. C₄²×D₅₃	C₂²	semidirect
43. C₂×#41×C₅₃	C₂²	semidirect
44. C₄⋊Q₈×C₅₃	C₂²	semidirect
46. #124×C₅₃	C₂²	semidirect
47. #121×C₅₃	C₂²	semidirect
48. C₄×D₄×C₅₃	C₂²	semidirect
49. C₂×C₄²×C₅₃	C₂²	semidirect
115. D₂₁₂	#41	semidirect
150. #532	D₅₃	semidirect
155. C₅₃	#3481	semidirect
157. C₂×#41	C₂×D₅₃	semidirect
160. C₄⋊Q₈	C₂×D₅₃	semidirect
165. #124	C₂×D₅₃	semidirect
173. #121	C₂×D₅₃	semidirect
186. C₄×D₄	C₂×D₅₃	semidirect
193. C₂×C₄²	C₂×D₅₃	semidirect